[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-123458-en":3,"doc-seo-123458-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},123458,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Machine Learning with Physics Knowledge for Prediction - A Survey","This survey examines a broad set of methods and models that merge machine learning with physics knowledge for prediction and forecasting, with an emphasis on partial differential equations. The work highlights strong interest driven by the ability to improve scientific research and industrial practice through useful inductive biases. It presents two main avenues: encoding physics knowledge via architectures and objectives, and treating data itself as physics knowledge via multi-task, meta, and contextual learning, followed by an industrial perspective and discussion of open-source ecosystems.","arXiv :2408 .09840v2 [ cs .LG] 15 May 2025  \nMachine Learning with Physics Knowledge for Prediction: A Survey  \nJoe Watson 1 ,2 ,†, Chen Song6 , Oliver Weeger3 ,7 , Theo Gruner 1 ,3 , An T. Le 1 , Kay Pompetzki 1 , Ahmed Hendawy 1 ,3 , Oleg Arenz 1 , Will Trojak8 , Miles Cranmer9 , Carlo D’Eramo 1 ,3 ,4 , Fabian  \nBülow6 , Tanmay Goyal6 , Jan Peters 1 ,2 ,3 ,5 , Martin W. Hoffman6 {joe, theo, an, kay, ahmed, oleg, carlo, [jan}@robot-learning. de](jan}@robot-learning. de)[ ](jan}@robot-learning. de){[chen. song](chen. song) , martin. w.hoffmann, fabian. bue low, [tanmay.goya l}@de. abb. com](tanmay.goya l}@de. abb. com)[ ](tanmay.goya l}@de. abb. com)[weeger@cps. tu-darmstadt. de](weeger@cps. tu-darmstadt. de)  \n[w. trojak@ibm. com](w. trojak@ibm. com)  \n1 Department of Computer Science, Technical University of Darmstadt, Germany  \n2 Systems AI for Robot Learning, German Research Center for AI (DFKI), Germany  \n3 Hessian Center for Artificial Intelligence (hessian. AI), Germany  \n4 Center for Artificial Intelligence and Data Science, University of Würzburg, Germany  \n5 Centre for Cognitive Science, Technical University of Darmstadt, Germany  \n6 ABB Corporate Research Center, Mannheim, Germany  \n7 Department of Mechanical Engineering, Technical University of Darmstadt, Germany  \n8 IBM Research UKI, United Kingdom  \n9 Data Intensive Science, University of Cambridge, United Kingdon † Now at the Oxford Robotics Institute, University of Oxford  \nReviewed on OpenReview: [https: // openreview. net/ forum? id= ZiJYahyXLU](https: // openreview. net/ forum? id= ZiJYahyXLU)  \nAbstract  \nThis survey examines the broad suite of methods and models for combining machine learning with physics knowledge for prediction and forecasting, with a focus on partial differential equations. These methods have attracted significant interest because of their potential impact on the advancement of scientific research and industrial practices, promising improvements to using small-or large-scale datasets and expressive predictive models with useful inductive biases. The survey has two parts. The first considers incorporating physics knowledge on an architectural level through objective functions, structured predictive models, and data augmentation. The second considers data as physics knowledge, which motivates looking at multi-task, meta, and contextual learning as an alternative approach to incorporating physics knowledge in a data-driven fashion. Finally, we also provide an industrial perspective on the application of these methods and a survey of the open-source ecosystem for physics-informed machine learning.  \n1 Introduction  \nPrediction lies at the heart of science and engineering, and many advances involve the discovery of simple patterns—often expressed as symbolic expressions—that approximate the evolution of our physical world. Still, these mathematical models are only achieved through some degree of simplification and abstraction, and thus rarely capture the complexity of the physical system in its full fidelity. As a result, there is significant interest in learning models from only measurements of the real world, spurring the development of machine learning (ML) (Bishop and Nasrabadi, 2006) for the physical sciences. This field relies on using large datasets of measurements to constrain highly flexible parametric models rather than evoking domain knowledge.  \nMachine learning methods differ greatly in the small-and big-data regimes. The small data regime, popularized since the early days of machine learning, covering state estimation, system identification, and kernel  \nmethods, considers the setting where data is used to estimate a few unknown parameters given many assumptions (Chiuso and Pillonetto, 2019) . The big data era, popularized by deep learning, can leverage large-scale datasets to train over-parameterized models that learn their own internal representations of underlying systems (L’heureux et al. , 2017) . Fueled by the ever-increa","cbCaiotF4C1FbUBn","https://ap.wps.com/l/cbCaiotF4C1FbUBn","pdf",1694839,1,61,"English","en",105,"# Introduction\n## Prediction in science and engineering\n## Small-data vs big-data regimes\n## Why physics knowledge improves reliability\n## Inductive biases as the central challenge\n# Survey scope and structure\n## Architectural incorporation of physics knowledge\n## Data as physics knowledge: multi-task, meta, contextual learning\n## Industrial perspective and open-source ecosystem","[{\"question\":\"What does the survey focus on?\",\"answer\":\"The survey focuses on methods and models that combine machine learning with physics knowledge for prediction and forecasting, especially in settings involving partial differential equations.\"},{\"question\":\"How does the survey organize the approaches to incorporating physics knowledge?\",\"answer\":\"It divides approaches into two parts: architectural integration through objective functions, structured predictive models, and data augmentation, and an alternative view where data is treated as physics knowledge via multi-task, meta, and contextual learning.\"},{\"question\":\"Why are physics-informed models considered useful for prediction tasks?\",\"answer\":\"They aim to improve reliability, robustness, and trust in predictions by adding inductive biases derived from scientific understanding, which helps learning effectively even with limited data.\"}]","Machine Learning with Physics Knowledge for Prediction - 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